For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
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1answer
365 views

A trivial solution vs. a non-trivial solution - involving vectors

I'm not entirely sure I understood this question in my text book, but it said the following: The zero vector $0 = \left(0,0,0\right)$ can be written as a linear combination of the vectors $v_1$, ...
2
votes
1answer
72 views

If $A$ and $B$ are symmetric, $A^5=B^5$ then $A=B$

Let $A$, $B$ be real symmetric $n\times n$ matrices such that $A^5=B^5$. Prove that $A=B$. Here are my attempts: The following identity holds $(A-B)(A^4+A^3B+A^2B^2+AB^3+B^4)=0$ and yields at least ...
3
votes
1answer
123 views

How find this matrix the inverse $A^{-1}$

Let $a,b>0$,and the matrix $A_{n\times n}$ and such $$A=\begin{bmatrix} a&b&0&\cdots&0&0\\ b&a&b&\cdots&0&0\\ 0&b&a&\cdots&0&0\\ ...
1
vote
1answer
101 views

On Adjacency Matrix of a Graph with a Cut Vertex and a Bridge

Let $G$ be a graph. If $v_i$ (resp. $v_iv_j$) is a cut vertex (resp. a bridge) of $G$, what can you say about its adjacency matrix $A(G)$?
-1
votes
1answer
35 views

Inverse of a Matrix Transformation

Assuming we have a transformation T(x)=A*x what is the process for finding the inverse of the transformation T^(-1)(x)? For example take A=(1 2) (1 1)
0
votes
2answers
99 views

Rank and Invertibility Problem - Non Square Matrix

Let $A \in \mathcal M_{m×n}(F)$. Prove that if $\text{rank}(A) = m$, then there exists $B \in \mathcal M_{n×m}(F)$ such that $AB = I_m$. I think I need to prove that $A^{-1}$ exists, such that ...
1
vote
2answers
80 views

Prove that if $A=\frac{1}{2}\{B,A\}$ then $B=1\!\mathrm{l}$

So, manipulating some (square) matrices, $A$ and $B$, I encountered an equation of the form: $$A=\tfrac{1}{2}\{A,B\}$$ where "$\{\cdot,\cdot\}$" denotes the anticommutator between matrices $A$ and ...
1
vote
1answer
97 views

What is transition matrix

Every e_j har coordinates in the first base: $$e_j = \sum_{i} s_{ij}e_i $$ so writing those coordinates as column vectors we get an important transition matrix $S = (s_{ij})$ and Theorem: ...
0
votes
0answers
49 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
1
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0answers
27 views

Reversing a regular transformation matrix

I'm working in 2D, with 3x3 matrices. I have an object at position T. I want to rotate/scale around the origin. Origin position O Rotation R Scale S Origin position P To find my matrix, I would ...
0
votes
2answers
142 views

How do rotational matrices work? [duplicate]

I am confuse on the how exactly rotational matrices work. So I understand that you can rotate a point around the x, y and z axis but if asked how you find a single matrix that will show the same ...
0
votes
2answers
82 views

Rotation of Matrices and their interpretation

Given are now two matrices and I have to discuss what the given functions are doing (geometrically). Maybe you can revise/add the following: given are the matrices $ A = \begin{pmatrix} \cos(a) ...
2
votes
0answers
33 views

markov spectral radius independent of states?

Let $\Pi$ be a stochastic matrix of an irreducible markov chain. We define the spectral radius of $\Pi$ as: $\rho(\Pi) := \limsup_{n \to \infty} \left( \pi^{(n)}_{(a,b)} \right)^{\frac{1}{n}}$ Why ...
0
votes
1answer
61 views

Properties of a matrix and eigenvalues

A, B, C are three real-square matrices. A is an upper triangular matrix with all of its diagonal entries equal to zero. B is a matrix such that $b_{ij}=-b_{ji}$, and C is a matrix such that $\sum_j ...
1
vote
2answers
66 views

Linear Algebra - inverse of a complex matrix

This is from the UPenn prelim questions. http://hans.math.upenn.edu/amcs/AMCS/prelims/prelim_review.pdf (question 12 in Linear Algebra session) Let A be a real symmetric matrix and form the matrix ...
2
votes
1answer
77 views

Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
2
votes
2answers
19 views

Multiplication by elements of SL_2

I'm thinking about $SL_2(\mathbb{Z})$. Suppose we fix the $2\times1$ matrix $(q_0$ $1)^T$, for $q_0$ a fixed rational number. My question is: Is it possible to get any $2\times1$ matrix of the form ...
1
vote
0answers
20 views

H-polar decomposition

Can anyone help me understand how this paper: http://www.cs.vu.nl/~ran/polarnormal.pdf figured out $(14)$ from Example 12 (on page 14)? The problem is essentially asking for a polar decomposition of ...
0
votes
1answer
42 views

Symmetric rank two correction of a symmetric positive definite matrix

Let $P$ is a symmetric positive definite matrix and define $P'$ as follow: $$P'=P + (e_ie_j^T+e_je_i^t)P_{ij}$$ (where $i \neq j$). (so $P'$ is a rank 2, symmetric correction for $P$) Is $P'$ is ...
2
votes
1answer
63 views

Can't show that these matrices are diagonalizable.

Consider that for each $n \times n$ (possibly complex) matrix, $A_{k}$, $0 \leq k \leq m$, we have that \begin{align} A_{0}A_{k} &= kA_{k}, \qquad 1 \leq k \leq m \end{align} and suppose that ...
0
votes
1answer
395 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
0
votes
2answers
89 views

Constructing Matrix (Rotation, Reflection)

Construct the matrix corresponding to a rotation of 90 degrees about the y-axis together with a reflection about the (x,z) plane. Reviewing Linear Algebra and seem to have forgotten some stuff. ...
2
votes
1answer
89 views

Proof that two square, diagonal matrices A and B fulfill the first binomial formula

In the current exercise for linear algebra, we had to find conditions so that two arbitrary quadratic matrices A and B with the same dimension satisfy the first binomial formula: (A+B)^2 = A^2 + 2AB ...
1
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1answer
43 views

Question about Hermitian Symmetric matrix

Assume that $A$ is a Hermitian symmetric n x n matrix with complex entries having all of its eigenvalues lying inside the interval, (-1, 1). Is $A^3 + Id$ always a positive definite matrix? My hunch ...
0
votes
0answers
39 views

Inverse of a non-singular linear transformation

The question is about showing that if A is a non-singular linear transformation of an n-dimensional linear space to itself, then there must be some polynomial $c_0 + c_1 z + ... + c_k z^k$ such that ...
1
vote
0answers
51 views

“Other” one-sided inverses

If a matrix has linearly independent columns, then if it's a square matrix it has linearly independent rows. Forget square matrices; think of this: $$ A=\begin{bmatrix} \bullet & \bullet \\ ...
1
vote
0answers
28 views

Online tools for plotting and deriving formulas from set of data?

I have a matrix of 20x3 decimal values. I would like to have them in a graph, with one plot for each of the three y values. I would also like approximated formulas for each of the three. Are there any ...
1
vote
1answer
48 views

Commutative matrices of order 2

I am trying to give a simple proof of why $GL_n(K)$ spaces are not isomorphic for different $n$, by finding $2^n$ (I presume this is the maximal possible number) of pairwise commutative matrices who ...
2
votes
1answer
130 views

Product of 3 Matrices

Is this identity true for all possible values of $A,B$ and $C$? $$A^TBC = C^TBA$$ where either $A,B,C$ are square matrices of same size $A$ and $C$ are vectors of size $n$ and $B$ is square matrix ...
2
votes
1answer
63 views

How to frame this set of linear equations?

I have the following set of equations, as an example $2x + 1y + 2z = A$ $0x + 2y + 2z = A$ $1x + 2y + 1z = A$ I assume this can be rewritten as a matrix? How can I check if a solution exists such ...
1
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1answer
87 views

how did binet get to his matrix multiplication algorithm and where does he speak about it?

Well I think it's all in the title..anyway..I would like to know through which passages did Binet get to his matrix multiplication algorithm and where (book, paper..) did he first tell about it to the ...
2
votes
1answer
65 views

How to find $A=UDU^H$ in this case

I am given a matrix $A$. I find out it is normal. And I compute $\det(A-\lambda)=0$ and find that not all $\lambda_i$ are different, i.e., the eigenvalues are not distinct. Thus, I am not sure if the ...
0
votes
1answer
44 views

Simple Matrix Caluculation

I am new to Math, and my knowledge is not good. Can anyone help me with this problem? Show that $(M^*)^* = M$, $(M^T)^T = M$, and $(M^t)^t = M$. I think I got some idea on the last 2, but the I am ...
2
votes
3answers
106 views

Proving that $\lambda$ being an eigenvalue for $A$ implies $\lambda^{-1}$ is an eigenvalue for $A^{-1}$

Let $A$ be an invertible matrix, and let $\lambda$ be an eigenvalue for $A$. We have that $Ax = \lambda x$ for some eigenvector $x$. Note that $A^{-1}Ax = A^{-1}\lambda x$, which gives $x = ...
2
votes
2answers
125 views

diagonalize a non-normal matrix , without distinct eigenvalues

I wonder how to diagonalize a matrix that is non-normal, and does not have distinct eigenvalues. Let $\lambda_i$ be the eigenvalue, and $v_i$ be the eigenvector with that eigenvalue. I think the ...
4
votes
1answer
394 views

Cramer's Rule Question

Use Cramer's rule to solve this system for z: $$2x+y+z=1$$ $$3x+z=4$$ $$x-y-z=2$$ so my work is: $$\frac{\left|\begin{matrix} 2 & 1 & 1\\ 3 & 0 & 4\\ 1 & -1 & 2 ...
1
vote
1answer
97 views

Upper triangular matrix and nilpotent

How to show that an upper triangular matrix is nilpotent if and only if all its diagonal elements are equal to zero by using induction?
3
votes
1answer
120 views

If $A$ is normal such that $AB=BA$, then $A^*B=BA^*$

Please help me. Let $A,B\in M_n(\mathbb{C})$. Show that if $A$ is normal such that $AB=BA$, then $A^*B=BA^*$.
1
vote
1answer
56 views

Largest eigenvalue of a special m-matrix

How to estimate the largest eigenvalue of followed characteristics? Let $A={a_{ij}}$. Symmetric positive definite. Real. Very sparse. Diagonal elements are all positive, and off-diagonal elements ...
0
votes
1answer
19 views

Order: multiplying matrix by a scalar

Let's say we have some matrix A and a constant c. Does it generally hold that cA=Ac? Thanks
1
vote
1answer
89 views

Proving properties of linear maps on one-dimensional vectors

An exercise from the book "Linear Algebra Done Right" asks to prove the following: 'Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More ...
0
votes
1answer
68 views

Is it true that 2 matrices are similar **if and only if ** they have the same Jordan form?

Is it true that 2 matrices are similar if and only if they have the same Jordan form? I know that one direction is correct: if have the same Jordan form -> similar. Is the other direction correct - ...
0
votes
0answers
39 views

Linear transformations and bases

Let $V=R^3$ and let $T: V \to V$ be the linear transformation defined by $T(x,y,z)=(3x-z,2y+5z,-7x+4z)$ (a) Using standard basis $\mathscr{E} = {(1,0,0),(0,1,0),(0,0,1)}$ for $V$ find the ...
3
votes
1answer
260 views

Gershgorin Circle Theorem: counterexample to a statement in the proof?

I have been struggling to comprehend the proof of Gershgorin Circle Theorem for a long time now, but I think I have come upon a counterexample. I'm probably wrong, but please tell me where I'm ...
5
votes
3answers
74 views

A question on eigenvalues

Let $A,B\in M_{2}(\mathbb{R})$ so that $A^2 = B^2 = I$. Which are eigenvalues of $AB$? 1) $1\pm \sqrt 3$ 2)$3 \pm 2\sqrt2$ 3)$\dfrac {1}{2},2$ 4)$2 \pm 2\sqrt 3$
1
vote
0answers
66 views

When is a matrix diagonalizable?

The matrix $A$ is diagonalizable with $A=SDS^{-1}$ where $D$ is a diagonal matrix and $S$ is the matrix with eigenvectors as columns iff $A$ has linearly independent eigenvectors. Correct?
0
votes
1answer
59 views

Complexity of sparse back substitution

What is the complexity of sparse backsubstitution $Rx = b$, given $n$, the dimensions of dense $x$ and $b$ as well as of the sparse $R$ and $nnz$, the number of nonzero entries in $R$?
1
vote
1answer
156 views

Quaternion and Matrix

I have a quaternion for rotation and a matrix for changing axis(change coordinate from camera to my rendering scene ). I have tested two method and i except to have equal resuls but results are ...
2
votes
1answer
61 views

Why does matrix-matrix product come close to the peak performance of a system?

In this paper, I read " The most important operation is GEMM (GEneral Matrix Multiply), which typically defines the practical peak performance of a computer system." But why? Why does matrix-matrix ...
1
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0answers
143 views

Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...